UPPER BOUNDS ON PACKING DENSITY FOR CIRCULAR

CYLINDERS WITH HIGH ASPECT RATIO

WODEN KUSNERDEPARTMENT OF MATHEMATICS

UNIVERSITY OF PITTSBURGH

Abstract. In the early 1990s, A. Bezdek and W. Kuperberg used a relatively

simple argument to show a surprising result: The maximum packing density ofcircular cylinders of infinite length in R3 is exactly π/

√12, the planar packing

density of the circle. This paper modifies their method to prove a bound on

the packing density of finite length circular cylinders. In fact, the maximumpacking density for unit radius cylinders of length t in R3 is bounded above

by π/√

12 + 10/t.

1. Introduction

The problem of computing upper bounds for the packing density of a specificbody in R3 can be difficult. Some known or partially understood non-trivial classesof objects are based on spheres [4], bi-infinite circular cylinders [2], truncated rhom-bic dodecahedra [1] and tetrahedra [3]. This paper proves an upper bound for thepacking density of congruent capped circular cylinders in R3. The methods areelementary, but can be used to prove non-trivial upper bounds for packings bycongruent circular cylinders and related objects, as well as the sharp bound forhalf-infinite circular cylinders.

1.1. Synopsis. The density bound of A. Bezdek and W. Kuperberg for bi-infinitecylinders is proved in three steps. Given a packing of R3 by congruent bi-infinitecylinders, first decompose space into regions closer to a particular axis than to anyother. Such a region contains the associated cylinder, so density may be determinedwith respect to a generic region. Finally, this region can be sliced perpendicular tothe particular axis and the area of these slices estimated: This area is always largecompared to the cross-section of the cylinder.

In the case of a packing of R3 by congruent finite-length cylinders, this methodfails. The ends of a cylinder may force some slice of a region to have small area.For example, if a cylinder were to abut another, a region in the decompositionmight not even wholly contain a cylinder. To overcome this, one shows that thesepotentially small area slices are always associated to a small neighborhood of theend of a cylinder. For a packing by cylinders of a relatively high aspect ratio,neighborhoods of the end of a cylinder are relatively rare. By quantifying therarity of cylinder ends in a packing, and bounding the error contributed by anyparticular cylinder end, the upper density bound is obtained.

Research partially supported by NSF grant 1104102.

1

arX

iv:1

309.

6996

v1 [

mat

h.M

G]

26

Sep

2013

2 WODEN KUSNER

1.2. Objects considered. Define a t-cylinder to be a closed solid circular cylinderin R3 with unit radius and length t. Define a capped t-cylinder (Figure 1) to bea closed set in R3 composed of a t-cylinder with solid unit hemispherical caps. Acapped t-cylinder C decomposes into the t-cylinder body C0 and two caps C1 andC2. The axis of the capped t-cylinder C is the line segment of length t forming theaxis of C0. The capped t-cylinder C is also the set of points at most 1 unit from itsaxis.

1.3. Packings and densities. A packing of X ⊆ R3 by capped t-cylinders is acountable family C = {Ci}i∈I of congruent capped t-cylinders Ci with mutuallydisjoint interiors and Ci ⊆ X. For a packing C of R3, the restriction of C toX ⊆ R3 is defined to be a packing of R3 by capped t-cylinders {Ci : Ci ⊆ X}. LetB(R) be the closed ball of radius R centered at 0. In general, let Bx(R) be theclosed ball of radius R centered at x. The density ρ(C , R,R′) of a packing C of R3

by capped t-cylinders with R ≤ R′ is defined as

ρ(C , R,R′) =∑

Ci⊆B(R)

Vol(Ci)

Vol(B(R′)).

The upper density ρ+ of a packing C of R3 by capped t-cylinders is defined as

ρ+(C ) = lim supR→∞

ρ(C , R,R).

Figure 1. A capped t-cylinder with body C0, axis a and caps C1

and C2.

In general, a packing of X ⊆ R3 by a convex, compact object K is a countablefamily K = {Ki}i∈I of congruent copies of K with mutually disjoint interiors andKi ⊆ X. Restrictions and densities of packings by K are defined in an analogousfashion to those of packings by capped t-cylinders.

2. The Main Results

Let t0 = 43 ( 4√

3+1)3 = 48.3266786 . . . for the remainder of the paper. This value

comes out of the error analysis in Section 5.

3

Theorem 1. Fix t ≥ 2t0. Fix R ≥ 2/√

3. Fix a packing C of R3 by capped t-cylinders. Then

ρ(C , R− 2/√

3, R) ≤t+ 4

3√12π (t− 2t0) + (2t0) + 4

3

.

This is the content of Sections 3, 4, 5. Notice that for t ≤ 2t0, the only upperbound provided is the trivial one.

Corollary 2. Fix t ≥ 2t0. The upper density of a packing C of R3 by cappedt-cylinders satisfies the inequality

ρ+(C ) ≤t+ 4

3√12π (t− 2t0) + (2t0) + 4

3

.

Proof. Let VR and WR be subsets of the index set I, with VR = {i : Ci ⊆ B(R)}and WR = {i : Ci ⊆ B(R− 2/

√3)}. By definition,

ρ+(C ) = lim supR→∞

(∑WR

Vol(Ci)

Vol(B(R))+

∑VRrWR

Vol(Ci)

Vol(B(R))

).

As R grows, the term∑VRrWR

Vol(Ci)/Vol(B(R)) tends to 0. Further analysisof the right-hand side yields

ρ+(C ) = lim supR→∞

ρ(C , R− 2/√

3, R).

By Theorem 1, the stated inequality holds. �

Lemma 3. Given a packing of t-cylinders with density ρ where t is at least 2, there

is a packing of capped (t− 2)-cylinders with packing density (t− 2

3

t ) · ρ.

Figure 2. Nesting capped (t− 2)-cylinders in t-cylinders.

Proof. From the given packing of t-cylinders, construct a packing by capped (t−2)-cylinders by nesting as illustrated in Figure 2. By comparing volumes, this packingof capped (t− 2)-cylinders has the required density. �

4 WODEN KUSNER

Corollary 4. Fix t ≥ 2t0 + 2. The upper density of a packing Z of R3 by t-cylinders satisfies the inequality

ρ+(Z ) ≤ t√12π (t− 2− 2t0) + (2t0) + 4

3

.

Proof. Assume there exists a packing by t-cylinders exceeding the stated bound.Then Lemma 3 gives a packing of capped (t−2)-cylinders with density greater than

(t− 2

3

t

)·

(t

√12π (t− 2− 2t0) + (2t0) + 4

3

)=

(t− 2 + 4

3√12π (t− 2− 2t0) + (2t0) + 4

3

).

This contradicts the density bound of Theorem 1 for capped (t−2)-cylinders. �

3. Set Up

For the remainder of the paper, fix the notation C ∗ to be the restriction of C toB(R− 2/

√3), indexed by I∗. To bound the density ρ(C ∗, R− 2/

√3, R) for a fixed

packing C and a fixed R ≥ 2/√

3, decompose B(R) into regions Di with disjointinteriors such that Ci ⊆ Di for all i in I∗. For such a packing C ∗ with fixed R,define the Dirichlet cell Di of a capped t-cylinder Ci to be the set of points in B(R)no further from the axis ai of Ci than from any other axis aj of Cj .

For any point x on axis ai, define a plane Px normal to ai and containing x.Define the Dirichlet slice dx be the set Di∩Px. For a fixed Dirichlet slice dx, defineSx(r) to be the circle of radius r centered at x in the plane Px. Important circlesare Sx(1), which coincides with the cross section of the boundary of the cylinder,

and Sx(2/√

3), which circumscribes the regular hexagon in which Sx(1) is inscribed.An end of the capped t-cylinder Ci refers to an endpoint of the axis ai.

Figure 3. A capped cylinder C and the slab L.

Define the slab Li to be the closed region of R3 bounded by the normal planesto ai through the endpoints of ai and containing C0

i (Figure 3). The Dirichlet cellDi decomposes into the region D0

i = Di ∩ Li containing C0i and complementary

regions D1i and D2

i containing the caps C1i and C2

i respectively (Figure 4).

5

Figure 4. Decomposing a Dirichlet cell.

Aside from a few degenerate cases, the set of points equidistant from a point xand line segment a in the affine hull of x and a form a parabolic spline (Figure 5).A parabolic spline is a parabolic arc extending in a C1 fashion to rays at the pointsequidistant to both the point x and an endpoint of the line segment a. Call thepoints where the parabolic arc meets the rays the Type I points of the curve. Aparabolic spline cylinder is a surface that is the cylinder over a parabolic spline.

Figure 5. Parabolic spline associated with point x and segment aj .

4. Qualified Points

Definition 5. Fix a packing C of R3 by capped t-cylinders. Fix R ≥ 2/√

3 andrestrict to C ∗. A point x on an axis is qualified if the Dirichlet slice dx has areagreater than

√12, the area of the regular hexagon in which Sx(1) is inscribed.

Proposition 6. Fix a packing C of R3 by capped t-cylinders. Fix R ≥ 2/√

3 andrestrict to C ∗. Let x be a point on an axis ai, where i is a fixed element of I∗. IfBx(4/

√3) contains no ends of C∗, then x is qualified.

The proof of this proposition will modify the Main Lemma of [2]. A series oflemmas allow for the truncation and rearrangement of the Dirichlet slice. The goalis to construct from dx a subset d∗∗x of Px with the following properties:

6 WODEN KUSNER

• d∗∗x contains Sx(1).• The boundary of d∗∗x is composed of line segments and parabolic arcs with

apexes touching Sx(1).

• The non-analytic points of the boundary of d∗∗x lie on Sx(2/√

3).• The area of d∗∗x is less than the area of dx.

Then the computations of [2, §6] apply.

Lemma 7. If a point x satisfies the conditions of Proposition 6, then the Dirichletslice dx is a bounded convex planar region, the boundary of which is a simple closedcurve consisting of a finite union of parabolic arcs, line segments and circular arcs.

Proof. Without loss of generality, fix a point x on ai. For each j 6= i in I∗, let dj

be the set of points in Px no further from ai than from aj . The Dirichlet slice dxis the intersection of B(R) with dj for all j 6= i in I∗. The boundary of dj is the setof points in Px that are equidistant from ai and aj . As Px is perpendicular to aiat x, the boundary of dj is also the set of points in Px equidistant from x and aj .

This is the intersection of the plane Px with the set of points in R3 equidistantfrom x and aj . The set of points in R3 equidistant from x and aj is a parabolicspline cylinder perpendicular to the affine hull of x and aj . Therefore the set ofpoints equidistant from x and aj in Px is also a parabolic spline, with x on theconvex side.

In the degenerate cases where x is in the affine hull of aj or Px is parallel to aj ,the set of points equidistant from x and aj in Px are lines or is empty.

The region dx is clearly bounded as it is contained in B(R). The point x lies inthe convex side of the parabolic spline so each region dj is convex. The set B(R)contains x and is convex, so dx is convex. This is a finite intersection of regionsbounded by parabolic arcs, lines and a circle, so the rest of the lemma follows. �

To apply the results of [2], the non-analytic points of the boundary of the Dirich-let slice dx must be controlled. From the construction of dx as a finite intersection,the non-analytic points of the boundary of dx fall into three non-disjoint classes ofpoints: the Type I points of a parabolic spline that forms a boundary arc of dx,Type II points defined to be points on the boundary of dx that are also on theboundary of B(R), and Type III points, defined to be points on the boundary ofdx that are equidistant from three or more axes. Type III points are the points onthe boundary of dx where the parabolic spline boundaries of various dj intersect.

Lemma 8. If a point x satisfies the conditions of Proposition 6, then no non-analytic points of the boundary of dx are in int(Conv(Sx(2/

√3))), where the interior

is with respect to the subspace topology of Px and Conv(·) is the convex hull.

Proof. It is enough to show there are no Type I, Type II, or Type III points inint(Conv(Sx(2/

√3))). By hypothesis, Bx(4/

√3) contains no ends. The Type I

points are equidistant from x and an end. As there are no ends contained inBx(4/

√3), there are no Type I points in int(Conv(Sx(2/

√3))).

By hypothesis, x is in B(R − 2/√

3). Therefore there are no points on the

boundary of B(R) in int(Conv(Sx(2/√

3))) and therefore no Type II points in

int(Conv(Sx(2/√

3))).As a Type III point is equidistant from three or more axes, at some distance

`, it is the center of a ball tangent to three unit balls. This is because a cappedt-cylinder contains a unit ball which meets the ball of radius ` centered at the

7

Type III point. These balls do not overlap as the interiors of the capped t-cylindershave empty intersection. Lemma 3 of [6] states that if a ball of radius ` intersects

three non-overlapping unit balls in R3, then ` ≥ 2/√

3− 1. It follows that there are

no Type III points in int(Conv(Sx(2/√

3))). �

Lemma 9. Fix a packing C . Then for for all i 6= j and i, j ∈ I∗, there is asupporting hyperplane Q of int(Ci) that is parallel to ai and separating int(Ci ∩Li)from int(Cj ∩ Li).

Proof. We can extend Ci ∩Li to an infinite cylinder Ci where Cj ∩Li and Ci havedisjoint interiors. The sets Cj ∩Li and Ci are convex, so the Minkowski hyperplaneseparation theorem gives the existence of a hyperplane separating int(Cj ∩Li) andint(Ci). This hyperplane is parallel to the axis ai by construction. We may takeQ to be the parallel translation to a supporting hyperplane of int(Ci) that stillseparates int(Ci ∩ Li) from int(Cj ∩ Li). See Figure 6 for an example. �

Figure 6. The hyperplane Q separates int(Ci ∩ Li) from int(Cj ∩ Li).

Lemma 10. Fix a packing C . Fix a point x on the axis ai of Ci such that Bx(4/√

3)

contains no ends. Let y and z be points on the circle Sx(2/√

3). If each of y andz is equidistant from Ci and Cj, then the angle yxz is smaller than or equal to

2 arccos(√

3− 1) := α0, which is approximately 85.88◦.

Proof. By hypothesis, Bx(4/√

3) contains no ends, including the end of the axis ai.

Therefore any points of Cj that are not in Li are at a distance greater than 4/√

3from x. The points of Ci and Cj that y and z are equidistant from must be in theslab Li, so it is enough to consider y and z equidistant from Ci and Cj ∩ Li.

By construction, the hyperplane Q separates all points of Cj ∩ Li from x. Letk be the line of intersection between Px and Q. As y and z are at a distance of2/√

3− 1 from both Cj ∩ Li and Ci, they are at most that distance from Q. Theyare also at most that distance from k. The largest possible angle yxz occurs wheny and z are on the x side of k in Px, each at exactly the distance 2/

√3− 1 from k

as illustrated in Figure 7 . This angle is exactly 2 arccos(√

3− 1) := α0. �

The following lemma is proved in [2].

8 WODEN KUSNER

Figure 7. An extremal configuration for the angle α0.

Lemma 11. Let y and z be points on Sx(2/√

3) such that 60◦ < yxz < α0. Forevery parabola p passing through y and z and having Sx(1) on its convex side, letxypzx denote the region bounded by segments xy, xz, and the parabola p. Let p0denote the parabola passing through y and z and tangent to Sx(1) at its apex.

Area(xyp0zx) ≤ Area(xypzx).

4.1. Truncating and rearranging. Consider the Dirichlet slice dx of a point xsatisfying the conditions of Proposition 6. The following steps produce a regionwith no greater area than that of dx.

Step 1 : The boundary of dx intersects Sx(2/√

3) in a finite number of points.

Label them y1, y2, . . . yn, yn+1 = y1 in clockwise order. Intersect dx with Sx(2/√

3)and call the new region d∗x.

By Lemma 8, this is a region bounded by arcs of Sx(2/√

3), parabolic arcs and

line segments, with non-analytic points on Sx(2/√

3).Step 2 : For i = 1, 2, . . . , n if yixyi+1 > 60◦ and if the boundary of d∗x be-

tween yi and yi+1 is a circular arc of Sx(2/√

3), then introduce additional verticeszi1 , zi2 , . . . zik on the circular arc yiyi+1 so that the polygonal arc yizi1zi2 . . . zikyi+1

does not intersect Sx(1). Relabel the set of vertices v1, v2, . . . vm, vm+1 = v1 inclockwise order.

Step 3 : If vixvi+1 ≤ 60◦ then truncate d∗x along the line segment vivi+1 keepingthe part of d∗x which contains Sx(1). This does not increase area by construction.If vixvi+1 > 60◦ then vivi+1 is a parabolic arc. Replace it by the parabolic arcthrough vi and vi+1 touching Sx(1) at its apex. This does not increase area byLemma 10. This new region d∗∗x has smaller area than dx, contains Sx(1), andbounded by line segments and parabolic arcs touching Sx(1) at their apexes, with

all non-analytic points of the boundary on Sx(2/√

3). If consecutive non-analyticpoints on the boundary have interior angle no greater than 60◦, they are joined byline segments. If they have interior angle between 60◦ and α0, they are joined by aparabolic arc.

The following lemma is a consequence of [2, §6], which determines the minimumarea of such a region.

Lemma 12. The region d∗∗x has area at least√

12.

Proposition 6 follows.

9

5. Decomposition of B(R) and Density Calculation

5.1. Decomposition. Fix a packing C . Fix R ≥ 2/√

3 and restrict to C ∗. Let theset A be the union of the axes ai over I∗. Let dµ be the 1-dimensional Hausdorffmeasure on A. Let X be the subset of qualified points of A. Let Y be the subsetof A given by {x ∈ A : Bx( 4√

3) contains no ends}. Let Z be the subset of A given

by {x ∈ A : Bx( 4√3) contains an end}. Notice that Y ⊆ X ⊆ A from Proposition 6

and Z = A− Y by definition.The sets are A, X, Y , and Z are measurable. The set A is just a finite disjoint

union of lines in R3. The area of the Dirichlet slice dx is piecewise continuous on A,so X is a Borel subset of A. Similarly the conditions defining Y and Z make themBorel subsets of A. The ball B(R) is finite volume, so I∗ has some finite cardinalityn.

Decompose B(R) into the regions {D0i }ni=1, {D1

i }ni=1 and {D2i }ni=1. Further

decompose the regions {D0i }ni=1 into Dirichlet slices dx, where x is an element of A.

5.2. Density computation. From the definition of density,

ρ(C , R− 2/√

3, R) =

∑I∗ Vol(C0

i ) +∑I∗ Vol(C1

i ) +∑I∗ Vol(C2

i )∑I∗ Vol(D0

i ) +∑I∗ Vol(D1

i ) +∑I∗ Vol(D2

i )

as Cji ⊆ Dji , and Vol(C0

i ) = tπ, and Vol(C1i ) = Vol(C2

i ) = 23π, it follows that

(13) ρ(C , R− 2/√

3, R) ≤ntπ + n 4

3π∑I∗ Vol(D0

i ) + n 43π.

Finally, ρ(C , R− 2/√

3, R) is explicitly bounded by the following lemma.

Lemma 14. For t ≥ 2t0,∑I∗

Vol(D0i ) ≥ n(

√12(t− 2t0) + π(2t0)).

Proof. The sum∑I∗Vol(D0

i ) may be written as an integral of the area of theDirichlet slices dx over A ∑

I∗

Vol(D0i ) =

∫A

Area(dx) dµ.

Using the area estimates from Proposition 6, there is an inequality∫A

Area(dx) dµ ≥∫X

√12 dµ+

∫A−X

π dµ.

As√

12 > π and the integration is over a region A with µ(A) < ∞, passing tothe subset Y ⊆ X gives∫

X

√12 dµ+

∫A−X

π dµ ≥∫Y

√12 dµ+

∫A−Y

π dµ =

∫A−Z

√12 dµ+

∫Z

π dµ.

The measure of Z is the measure of the subset of A that is contained in all theballs of radius 4/

√3 about all the ends of all the cylinders in the packing. This

is bounded from above by considering the volume of cylinders contained in balls

10 WODEN KUSNER

of radius 4/√

3 + 1. If the cylinders completely filled the ball, they would containat most axis length 4

3 ( 4√3

+ 1)3 = t0. As each cylinder has two ends, there are at

worst 2n disjoint balls to consider. Therefore 2nt0 ≥ µ(Z).Provided t ≥ 2t0, we have the inequality∫

A−Z

√12 dµ+

∫Z

π dµ ≥ (nt− 2nt0)√

12 + 2n(t0)π.

�

By combining inequality (13) with Lemma 14 and simplifying, it follows that

ρ(C , R− 2/√

3, R) ≤t+ 4

3√12π (t− 2t0) + (2t0) + 4

3

for an arbitrary packing C of R3 by capped congruent t-cylinders.

6. Conclusions, applications, further questions

6.1. A rule of thumb, a dominating hyperbola. For t ≥ 0, the upper boundsfor the density of packings by capped and uncapped t-cylinders are dominated bycurves of the form π√

12+N/t. Numerically, one finds a useful rule of thumb:

Theorem 15. The upper density ρ+ of a packing C of R3 by capped t-cylinderssatisfies

ρ+(C ) ≤ π√12

+10

t.

Theorem 16. The upper density ρ+ of a packing C of R3 by t-cylinders satisfies

ρ+(C ) ≤ π√12

+10

t.

6.2. Examples. While the requirement that t be greater than 2t0 for a non-trivialupper bound is inconvenient, the upper bound converges rapidly to π/

√12 =

0.9069 . . . and is nontrivial for tangible objects, as illustrated in the table below.

Item Length Diameter t Density ≤

Broomstick 1371.6mm 25.4mm 108 0.9956...

20’ PVC Pipe 6096mm 38.1mm 320 0.9353...

Capellini 300mm 1mm 600 0.9219...

Carbon Nanotube - - 2.64× 108 [8] 0.9069...

6.3. Some further results. There are other conclusions to be drawn. For exam-ple: Consider the density of a packing C = {Ci}i∈I of R3 by congruent unit radiuscircular cylinders Ci, possibly of infinite length. The upper density γ+ of such apacking may be written

γ+(C ) = lim supr→∞

∑I

Vol(Ci ∩B0(r))

Vol(B0(r))

and coincides with ρ+(C ) when the lengths of Ci are uniformly bounded.

11

Theorem 17. The upper density γ of half-infinite cylinders is exactly π/√

12.

Figure 8. Packing cylinders with high density.

Proof. The lower bound is given by the obvious packing C ′ with parallel axes(Figure 8) and γ+(C ′) = π/

√12. Since a packing C (∞) of R3 by half-infinite

cylinders also gives a packing C (t) of R3 by t-cylinders, the inequality

t√12π (t− 2− 2t0) + (2t0) + 4

3

≥ ρ+(C (t)) = γ+(C (t)) ≥ γ+(C (∞))

holds for all t ≥ 2t0. �

Theorem 18. Given a packing C = {Ci}i∈I by non-congruent capped unit cylin-ders with lengths constrained to be between 2t0 and some uniform upper bound M ,the density satisfies the inequality

ρ+(C ) ≤t+ 4

3√12π (t− 2t0) + (2t0) + 4

3

where t is the average cylinder length given by lim infr→∞ µ(ai)/|J |, where J is theset {i ∈ I : Ci ⊆ B(r)}.

Proof. None of the qualification conditions require a uniform length t. Inequality13 may be considered with respect to the total length of A rather than nt. �

It may be easier to compute a bound using the following

Corollary 19. Given a packing C = {Ci}i∈I by non-congruent capped unit cylin-ders with lengths constrained to be between 2t0 and some uniform upper bound M ,the density satisfies the inequality

ρ+(C ) ≤t+ 4

3√12π (t− 2t0) + (2t0) + 4

3

where t is the infimum of cylinder length.

Proof. The stated bound is a decreasing function in t, so this follows from theprevious theorem. �

12 WODEN KUSNER

6.4. Questions and Remarks. Similar but much weaker results can be obtainedfor the packing density of curved tubes by realizing them as containers for cylinders.Better bounds on tubes would come from better bounds on t-cylinders for t small.There is the conjecture of Wilker, where the expected packing density of congruentunit radius circular cylinders of any length is exactly the planar packing densityof the circle, but that is certainly beyond the techniques of this paper. A moretractable extension of this might be to parametrize the densities for capped t-cylinders from the sphere to the infinite cylinder by controlling the ends. In thispaper, the analysis assumes anything in a neighborhood of an end packs with density1, whereas it is expected that the ends and nearby sections of tubes would packwith density closer to π/

√18. In [7], it is conjectured that the densest packing is

obtained from extending a dense sphere packing, giving a density bound of

ρ+(C (t)) =π√12

t+ 43

t+ 2√6

3

.

The structure of high density cylinder packings is also unclear. For infinite cir-cular cylinders, there are nonparallel packings with positive density [5]. In the caseof finite length t-cylinders, there exist nonparallel packings with density close toπ/√

12, obtained by laminating large uniform cubes packed with parallel cylinders,shrinking the cylinders and perturbing their axes. It is unclear how or if the align-ment of cylinders is correlated with density. Finally, as the upper bound presentedis not sharp, it is still not useable to control the defects of packings achieving themaximal density. A conjecture is that, for a packing of R3 by t-cylinders to achievea density of π/

√12, the packing must contain arbitrarily large regions of t-cylinders

with axes arbitrarily close to parallel.

References

[1] Andras Bezdek. A remark on the packing density in the 3-space. Intuitive Geometry, 63:17,

1994.[2] Andras Bezdek and Wlodzimierz Kuperberg. Maximum density space packing with congruent

circular cylinders of infinite length. Mathematika, 37(1):74–80, 1990.

[3] Simon Gravel, Veit Elser, and Yoav Kallus. Upper bound on the packing density of regulartetrahedra and octahedra. Discrete & Computational Geometry, 46(4):799–818, 2011.

[4] Thomas C. Hales. A proof of the Kepler conjecture. Annals of Mathematics, 162(3):1065–1185,2005.

[5] Krystyna Kuperberg. A nonparallel cylinder packing with positive density. Mathematika,

37(02):324–331, 1990.[6] Wlodzimierz Kuperberg. Placing and moving spheres in the gaps of a cylinder packing. Ele-

mente der Mathematik, 46(2):47–51, 1991.[7] Salvatore Torquato and Yang Jiao. Organizing principles for dense packings of nonspherical

hard particles: Not all shapes are created equal. Physical Review E, 86(1):011102, 2012.

[8] Xueshen Wang, Qunqing Li, Jing Xie, Zhong Jin, Jinyong Wang, Yan Li, Kaili Jiang, and

Shoushan Fan. Fabrication of ultralong and electrically uniform single-walled carbon nanotubeson clean substrates. Nano Letters, 9(9):3137–3141, 2009.